The proposed equivalence provides an alternative approach to establish contraction rates. In particular, rather than rely on the concentration function, one can resort to the convergence of the posterior mean and some mild conditions regarding the eigendecomposition of the covariance kernel of the GP prior, which are both systematically addressed in a unified non-asymptotic analysis (see our third contribution below) and an equivalent kernel technique. Hence, one may build on the well established literature on convergence rates of Bayes estimators, particularly through kernel ridge regression (KRR) [1]}, [2]}, [3]}, to conveniently obtain posterior contraction rates under both the \(L_2\)  and \(L_{\infty }\)  norms. As an example, under the proposed framework, we establish posterior contraction rates for Hölder smooth functions as well as analytic-type functions, which are minimax optimal up to a logarithmic factor.
